Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x-3y &= 5 \\ 3x+9y &= -9\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $1$ $\begin{align*}-15x-9y &= 15\\ 3x+9y &= -9\end{align*}$ Add the top and bottom equations. $-12x = 6$ Divide both sides by $-12$ and reduce as necessary. $x = -\dfrac{1}{2}$ Substitute $-\dfrac{1}{2}$ for $x$ in the top equation. $-5( -\dfrac{1}{2})-3y = 5$ $\dfrac{5}{2}-3y = 5$ $-3y = \dfrac{5}{2}$ $y = -\dfrac{5}{6}$ The solution is $\enspace x = -\dfrac{1}{2}, \enspace y = -\dfrac{5}{6}$.